// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2015 Eugene Brevdo <ebrevdo@gmail.com>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_SPECIAL_FUNCTIONS_H
#define EIGEN_SPECIAL_FUNCTIONS_H

namespace Eigen {
namespace internal {

    //  Parts of this code are based on the Cephes Math Library.
    //
    //  Cephes Math Library Release 2.8:  June, 2000
    //  Copyright 1984, 1987, 1992, 2000 by Stephen L. Moshier
    //
    //  Permission has been kindly provided by the original author
    //  to incorporate the Cephes software into the Eigen codebase:
    //
    //    From: Stephen Moshier
    //    To: Eugene Brevdo
    //    Subject: Re: Permission to wrap several cephes functions in Eigen
    //
    //    Hello Eugene,
    //
    //    Thank you for writing.
    //
    //    If your licensing is similar to BSD, the formal way that has been
    //    handled is simply to add a statement to the effect that you are incorporating
    //    the Cephes software by permission of the author.
    //
    //    Good luck with your project,
    //    Steve

    /****************************************************************************
 * Implementation of lgamma, requires C++11/C99                             *
 ****************************************************************************/

    template <typename Scalar> struct lgamma_impl
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE Scalar run(const Scalar)
        {
            EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false), THIS_TYPE_IS_NOT_SUPPORTED);
            return Scalar(0);
        }
    };

    template <typename Scalar> struct lgamma_retval
    {
        typedef Scalar type;
    };

#if EIGEN_HAS_C99_MATH
// Since glibc 2.19
#if defined(__GLIBC__) && ((__GLIBC__ >= 2 && __GLIBC_MINOR__ >= 19) || __GLIBC__ > 2) && \
    (defined(_DEFAULT_SOURCE) || defined(_BSD_SOURCE) || defined(_SVID_SOURCE))
#define EIGEN_HAS_LGAMMA_R
#endif

// Glibc versions before 2.19
#if defined(__GLIBC__) && ((__GLIBC__ == 2 && __GLIBC_MINOR__ < 19) || __GLIBC__ < 2) && (defined(_BSD_SOURCE) || defined(_SVID_SOURCE))
#define EIGEN_HAS_LGAMMA_R
#endif

    template <> struct lgamma_impl<float>
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE float run(float x)
        {
#if !defined(EIGEN_GPU_COMPILE_PHASE) && defined(EIGEN_HAS_LGAMMA_R) && !defined(__APPLE__)
            int dummy;
            return ::lgammaf_r(x, &dummy);
#elif defined(SYCL_DEVICE_ONLY)
            return cl::sycl::lgamma(x);
#else
            return ::lgammaf(x);
#endif
        }
    };

    template <> struct lgamma_impl<double>
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE double run(double x)
        {
#if !defined(EIGEN_GPU_COMPILE_PHASE) && defined(EIGEN_HAS_LGAMMA_R) && !defined(__APPLE__)
            int dummy;
            return ::lgamma_r(x, &dummy);
#elif defined(SYCL_DEVICE_ONLY)
            return cl::sycl::lgamma(x);
#else
            return ::lgamma(x);
#endif
        }
    };

#undef EIGEN_HAS_LGAMMA_R
#endif

    /****************************************************************************
 * Implementation of digamma (psi), based on Cephes                         *
 ****************************************************************************/

    template <typename Scalar> struct digamma_retval
    {
        typedef Scalar type;
    };

    /*
 *
 * Polynomial evaluation helper for the Psi (digamma) function.
 *
 * digamma_impl_maybe_poly::run(s) evaluates the asymptotic Psi expansion for
 * input Scalar s, assuming s is above 10.0.
 *
 * If s is above a certain threshold for the given Scalar type, zero
 * is returned.  Otherwise the polynomial is evaluated with enough
 * coefficients for results matching Scalar machine precision.
 *
 *
 */
    template <typename Scalar> struct digamma_impl_maybe_poly
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE Scalar run(const Scalar)
        {
            EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false), THIS_TYPE_IS_NOT_SUPPORTED);
            return Scalar(0);
        }
    };

    template <> struct digamma_impl_maybe_poly<float>
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE float run(const float s)
        {
            const float A[] = {-4.16666666666666666667E-3f, 3.96825396825396825397E-3f, -8.33333333333333333333E-3f, 8.33333333333333333333E-2f};

            float z;
            if (s < 1.0e8f)
            {
                z = 1.0f / (s * s);
                return z * internal::ppolevl<float, 3>::run(z, A);
            }
            else
                return 0.0f;
        }
    };

    template <> struct digamma_impl_maybe_poly<double>
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE double run(const double s)
        {
            const double A[] = {8.33333333333333333333E-2,
                                -2.10927960927960927961E-2,
                                7.57575757575757575758E-3,
                                -4.16666666666666666667E-3,
                                3.96825396825396825397E-3,
                                -8.33333333333333333333E-3,
                                8.33333333333333333333E-2};

            double z;
            if (s < 1.0e17)
            {
                z = 1.0 / (s * s);
                return z * internal::ppolevl<double, 6>::run(z, A);
            }
            else
                return 0.0;
        }
    };

    template <typename Scalar> struct digamma_impl
    {
        EIGEN_DEVICE_FUNC
        static Scalar run(Scalar x)
        {
            /*
     *
     *     Psi (digamma) function (modified for Eigen)
     *
     *
     * SYNOPSIS:
     *
     * double x, y, psi();
     *
     * y = psi( x );
     *
     *
     * DESCRIPTION:
     *
     *              d      -
     *   psi(x)  =  -- ln | (x)
     *              dx
     *
     * is the logarithmic derivative of the gamma function.
     * For integer x,
     *                   n-1
     *                    -
     * psi(n) = -EUL  +   >  1/k.
     *                    -
     *                   k=1
     *
     * If x is negative, it is transformed to a positive argument by the
     * reflection formula  psi(1-x) = psi(x) + pi cot(pi x).
     * For general positive x, the argument is made greater than 10
     * using the recurrence  psi(x+1) = psi(x) + 1/x.
     * Then the following asymptotic expansion is applied:
     *
     *                           inf.   B
     *                            -      2k
     * psi(x) = log(x) - 1/2x -   >   -------
     *                            -        2k
     *                           k=1   2k x
     *
     * where the B2k are Bernoulli numbers.
     *
     * ACCURACY (float):
     *    Relative error (except absolute when |psi| < 1):
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      0,30        30000       1.3e-15     1.4e-16
     *    IEEE      -30,0       40000       1.5e-15     2.2e-16
     *
     * ACCURACY (double):
     *    Absolute error,  relative when |psi| > 1 :
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      -33,0        30000      8.2e-7      1.2e-7
     *    IEEE      0,33        100000      7.3e-7      7.7e-8
     *
     * ERROR MESSAGES:
     *     message         condition      value returned
     * psi singularity    x integer <=0      INFINITY
     */

            Scalar p, q, nz, s, w, y;
            bool negative = false;

            const Scalar nan = NumTraits<Scalar>::quiet_NaN();
            const Scalar m_pi = Scalar(EIGEN_PI);

            const Scalar zero = Scalar(0);
            const Scalar one = Scalar(1);
            const Scalar half = Scalar(0.5);
            nz = zero;

            if (x <= zero)
            {
                negative = true;
                q = x;
                p = numext::floor(q);
                if (p == q)
                {
                    return nan;
                }
                /* Remove the zeros of tan(m_pi x)
       * by subtracting the nearest integer from x
       */
                nz = q - p;
                if (nz != half)
                {
                    if (nz > half)
                    {
                        p += one;
                        nz = q - p;
                    }
                    nz = m_pi / numext::tan(m_pi * nz);
                }
                else
                {
                    nz = zero;
                }
                x = one - x;
            }

            /* use the recurrence psi(x+1) = psi(x) + 1/x. */
            s = x;
            w = zero;
            while (s < Scalar(10))
            {
                w += one / s;
                s += one;
            }

            y = digamma_impl_maybe_poly<Scalar>::run(s);

            y = numext::log(s) - (half / s) - y - w;

            return (negative) ? y - nz : y;
        }
    };

    /****************************************************************************
 * Implementation of erf, requires C++11/C99                                *
 ****************************************************************************/

    /** \internal \returns the error function of \a a (coeff-wise)
    Doesn't do anything fancy, just a 13/8-degree rational interpolant which
    is accurate up to a couple of ulp in the range [-4, 4], outside of which
    fl(erf(x)) = +/-1.

    This implementation works on both scalars and Ts.
*/
    template <typename T> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE T generic_fast_erf_float(const T& a_x)
    {
        // Clamp the inputs to the range [-4, 4] since anything outside
        // this range is +/-1.0f in single-precision.
        const T plus_4 = pset1<T>(4.f);
        const T minus_4 = pset1<T>(-4.f);
        const T x = pmax(pmin(a_x, plus_4), minus_4);
        // The monomial coefficients of the numerator polynomial (odd).
        const T alpha_1 = pset1<T>(-1.60960333262415e-02f);
        const T alpha_3 = pset1<T>(-2.95459980854025e-03f);
        const T alpha_5 = pset1<T>(-7.34990630326855e-04f);
        const T alpha_7 = pset1<T>(-5.69250639462346e-05f);
        const T alpha_9 = pset1<T>(-2.10102402082508e-06f);
        const T alpha_11 = pset1<T>(2.77068142495902e-08f);
        const T alpha_13 = pset1<T>(-2.72614225801306e-10f);

        // The monomial coefficients of the denominator polynomial (even).
        const T beta_0 = pset1<T>(-1.42647390514189e-02f);
        const T beta_2 = pset1<T>(-7.37332916720468e-03f);
        const T beta_4 = pset1<T>(-1.68282697438203e-03f);
        const T beta_6 = pset1<T>(-2.13374055278905e-04f);
        const T beta_8 = pset1<T>(-1.45660718464996e-05f);

        // Since the polynomials are odd/even, we need x^2.
        const T x2 = pmul(x, x);

        // Evaluate the numerator polynomial p.
        T p = pmadd(x2, alpha_13, alpha_11);
        p = pmadd(x2, p, alpha_9);
        p = pmadd(x2, p, alpha_7);
        p = pmadd(x2, p, alpha_5);
        p = pmadd(x2, p, alpha_3);
        p = pmadd(x2, p, alpha_1);
        p = pmul(x, p);

        // Evaluate the denominator polynomial p.
        T q = pmadd(x2, beta_8, beta_6);
        q = pmadd(x2, q, beta_4);
        q = pmadd(x2, q, beta_2);
        q = pmadd(x2, q, beta_0);

        // Divide the numerator by the denominator.
        return pdiv(p, q);
    }

    template <typename T> struct erf_impl
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE T run(const T& x) { return generic_fast_erf_float(x); }
    };

    template <typename Scalar> struct erf_retval
    {
        typedef Scalar type;
    };

#if EIGEN_HAS_C99_MATH
    template <> struct erf_impl<float>
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE float run(float x)
        {
#if defined(SYCL_DEVICE_ONLY)
            return cl::sycl::erf(x);
#else
            return generic_fast_erf_float(x);
#endif
        }
    };

    template <> struct erf_impl<double>
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE double run(double x)
        {
#if defined(SYCL_DEVICE_ONLY)
            return cl::sycl::erf(x);
#else
            return ::erf(x);
#endif
        }
    };
#endif  // EIGEN_HAS_C99_MATH

    /***************************************************************************
* Implementation of erfc, requires C++11/C99                               *
****************************************************************************/

    template <typename Scalar> struct erfc_impl
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE Scalar run(const Scalar)
        {
            EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false), THIS_TYPE_IS_NOT_SUPPORTED);
            return Scalar(0);
        }
    };

    template <typename Scalar> struct erfc_retval
    {
        typedef Scalar type;
    };

#if EIGEN_HAS_C99_MATH
    template <> struct erfc_impl<float>
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE float run(const float x)
        {
#if defined(SYCL_DEVICE_ONLY)
            return cl::sycl::erfc(x);
#else
            return ::erfcf(x);
#endif
        }
    };

    template <> struct erfc_impl<double>
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE double run(const double x)
        {
#if defined(SYCL_DEVICE_ONLY)
            return cl::sycl::erfc(x);
#else
            return ::erfc(x);
#endif
        }
    };
#endif  // EIGEN_HAS_C99_MATH

    /***************************************************************************
* Implementation of ndtri.                                                 *
****************************************************************************/

    /* Inverse of Normal distribution function (modified for Eigen).
 *
 *
 * SYNOPSIS:
 *
 * double x, y, ndtri();
 *
 * x = ndtri( y );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the argument, x, for which the area under the
 * Gaussian probability density function (integrated from
 * minus infinity to x) is equal to y.
 *
 *
 * For small arguments 0 < y < exp(-2), the program computes
 * z = sqrt( -2.0 * log(y) );  then the approximation is
 * x = z - log(z)/z  - (1/z) P(1/z) / Q(1/z).
 * There are two rational functions P/Q, one for 0 < y < exp(-32)
 * and the other for y up to exp(-2).  For larger arguments,
 * w = y - 0.5, and  x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain        # trials      peak         rms
 *    DEC      0.125, 1         5500       9.5e-17     2.1e-17
 *    DEC      6e-39, 0.135     3500       5.7e-17     1.3e-17
 *    IEEE     0.125, 1        20000       7.2e-16     1.3e-16
 *    IEEE     3e-308, 0.135   50000       4.6e-16     9.8e-17
 *
 *
 * ERROR MESSAGES:
 *
 *   message         condition    value returned
 * ndtri domain       x <= 0        -MAXNUM
 * ndtri domain       x >= 1         MAXNUM
 *
 */
    /*
   Cephes Math Library Release 2.2: June, 1992
   Copyright 1985, 1987, 1992 by Stephen L. Moshier
   Direct inquiries to 30 Frost Street, Cambridge, MA 02140
 */

    // TODO: Add a cheaper approximation for float.

    template <typename T> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE T flipsign(const T& should_flipsign, const T& x)
    {
        typedef typename unpacket_traits<T>::type Scalar;
        const T sign_mask = pset1<T>(Scalar(-0.0));
        T sign_bit = pand<T>(should_flipsign, sign_mask);
        return pxor<T>(sign_bit, x);
    }

    template <> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE double flipsign<double>(const double& should_flipsign, const double& x)
    {
        return should_flipsign == 0 ? x : -x;
    }

    template <> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE float flipsign<float>(const float& should_flipsign, const float& x)
    {
        return should_flipsign == 0 ? x : -x;
    }

    // We split this computation in to two so that in the scalar path
    // only one branch is evaluated (due to our template specialization of pselect
    // being an if statement.)

    template <typename T, typename ScalarType> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE T generic_ndtri_gt_exp_neg_two(const T& b)
    {
        const ScalarType p0[] = {ScalarType(-5.99633501014107895267e1),
                                 ScalarType(9.80010754185999661536e1),
                                 ScalarType(-5.66762857469070293439e1),
                                 ScalarType(1.39312609387279679503e1),
                                 ScalarType(-1.23916583867381258016e0)};
        const ScalarType q0[] = {ScalarType(1.0),
                                 ScalarType(1.95448858338141759834e0),
                                 ScalarType(4.67627912898881538453e0),
                                 ScalarType(8.63602421390890590575e1),
                                 ScalarType(-2.25462687854119370527e2),
                                 ScalarType(2.00260212380060660359e2),
                                 ScalarType(-8.20372256168333339912e1),
                                 ScalarType(1.59056225126211695515e1),
                                 ScalarType(-1.18331621121330003142e0)};
        const T sqrt2pi = pset1<T>(ScalarType(2.50662827463100050242e0));
        const T half = pset1<T>(ScalarType(0.5));
        T c, c2, ndtri_gt_exp_neg_two;

        c = psub(b, half);
        c2 = pmul(c, c);
        ndtri_gt_exp_neg_two = pmadd(c, pmul(c2, pdiv(internal::ppolevl<T, 4>::run(c2, p0), internal::ppolevl<T, 8>::run(c2, q0))), c);
        return pmul(ndtri_gt_exp_neg_two, sqrt2pi);
    }

    template <typename T, typename ScalarType> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE T generic_ndtri_lt_exp_neg_two(const T& b, const T& should_flipsign)
    {
        /* Approximation for interval z = sqrt(-2 log a ) between 2 and 8
   * i.e., a between exp(-2) = .135 and exp(-32) = 1.27e-14.
   */
        const ScalarType p1[] = {ScalarType(4.05544892305962419923e0),
                                 ScalarType(3.15251094599893866154e1),
                                 ScalarType(5.71628192246421288162e1),
                                 ScalarType(4.40805073893200834700e1),
                                 ScalarType(1.46849561928858024014e1),
                                 ScalarType(2.18663306850790267539e0),
                                 ScalarType(-1.40256079171354495875e-1),
                                 ScalarType(-3.50424626827848203418e-2),
                                 ScalarType(-8.57456785154685413611e-4)};
        const ScalarType q1[] = {ScalarType(1.0),
                                 ScalarType(1.57799883256466749731e1),
                                 ScalarType(4.53907635128879210584e1),
                                 ScalarType(4.13172038254672030440e1),
                                 ScalarType(1.50425385692907503408e1),
                                 ScalarType(2.50464946208309415979e0),
                                 ScalarType(-1.42182922854787788574e-1),
                                 ScalarType(-3.80806407691578277194e-2),
                                 ScalarType(-9.33259480895457427372e-4)};
        /* Approximation for interval z = sqrt(-2 log a ) between 8 and 64
   * i.e., a between exp(-32) = 1.27e-14 and exp(-2048) = 3.67e-890.
   */
        const ScalarType p2[] = {ScalarType(3.23774891776946035970e0),
                                 ScalarType(6.91522889068984211695e0),
                                 ScalarType(3.93881025292474443415e0),
                                 ScalarType(1.33303460815807542389e0),
                                 ScalarType(2.01485389549179081538e-1),
                                 ScalarType(1.23716634817820021358e-2),
                                 ScalarType(3.01581553508235416007e-4),
                                 ScalarType(2.65806974686737550832e-6),
                                 ScalarType(6.23974539184983293730e-9)};
        const ScalarType q2[] = {ScalarType(1.0),
                                 ScalarType(6.02427039364742014255e0),
                                 ScalarType(3.67983563856160859403e0),
                                 ScalarType(1.37702099489081330271e0),
                                 ScalarType(2.16236993594496635890e-1),
                                 ScalarType(1.34204006088543189037e-2),
                                 ScalarType(3.28014464682127739104e-4),
                                 ScalarType(2.89247864745380683936e-6),
                                 ScalarType(6.79019408009981274425e-9)};
        const T eight = pset1<T>(ScalarType(8.0));
        const T one = pset1<T>(ScalarType(1));
        const T neg_two = pset1<T>(ScalarType(-2));
        T x, x0, x1, z;

        x = psqrt(pmul(neg_two, plog(b)));
        x0 = psub(x, pdiv(plog(x), x));
        z = pdiv(one, x);
        x1 = pmul(z,
                  pselect(pcmp_lt(x, eight),
                          pdiv(internal::ppolevl<T, 8>::run(z, p1), internal::ppolevl<T, 8>::run(z, q1)),
                          pdiv(internal::ppolevl<T, 8>::run(z, p2), internal::ppolevl<T, 8>::run(z, q2))));
        return flipsign(should_flipsign, psub(x0, x1));
    }

    template <typename T, typename ScalarType> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T generic_ndtri(const T& a)
    {
        const T maxnum = pset1<T>(NumTraits<ScalarType>::infinity());
        const T neg_maxnum = pset1<T>(-NumTraits<ScalarType>::infinity());

        const T zero = pset1<T>(ScalarType(0));
        const T one = pset1<T>(ScalarType(1));
        // exp(-2)
        const T exp_neg_two = pset1<T>(ScalarType(0.13533528323661269189));
        T b, ndtri, should_flipsign;

        should_flipsign = pcmp_le(a, psub(one, exp_neg_two));
        b = pselect(should_flipsign, a, psub(one, a));

        ndtri =
            pselect(pcmp_lt(exp_neg_two, b), generic_ndtri_gt_exp_neg_two<T, ScalarType>(b), generic_ndtri_lt_exp_neg_two<T, ScalarType>(b, should_flipsign));

        return pselect(pcmp_le(a, zero), neg_maxnum, pselect(pcmp_le(one, a), maxnum, ndtri));
    }

    template <typename Scalar> struct ndtri_retval
    {
        typedef Scalar type;
    };

#if !EIGEN_HAS_C99_MATH

    template <typename Scalar> struct ndtri_impl
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE Scalar run(const Scalar)
        {
            EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false), THIS_TYPE_IS_NOT_SUPPORTED);
            return Scalar(0);
        }
    };

#else

    template <typename Scalar> struct ndtri_impl
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE Scalar run(const Scalar x) { return generic_ndtri<Scalar, Scalar>(x); }
    };

#endif  // EIGEN_HAS_C99_MATH

    /**************************************************************************************************************
 * Implementation of igammac (complemented incomplete gamma integral), based on Cephes but requires C++11/C99 *
 **************************************************************************************************************/

    template <typename Scalar> struct igammac_retval
    {
        typedef Scalar type;
    };

    // NOTE: cephes_helper is also used to implement zeta
    template <typename Scalar> struct cephes_helper
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE Scalar machep()
        {
            assert(false && "machep not supported for this type");
            return 0.0;
        }
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE Scalar big()
        {
            assert(false && "big not supported for this type");
            return 0.0;
        }
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE Scalar biginv()
        {
            assert(false && "biginv not supported for this type");
            return 0.0;
        }
    };

    template <> struct cephes_helper<float>
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE float machep()
        {
            return NumTraits<float>::epsilon() / 2;  // 1.0 - machep == 1.0
        }
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE float big()
        {
            // use epsneg (1.0 - epsneg == 1.0)
            return 1.0f / (NumTraits<float>::epsilon() / 2);
        }
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE float biginv()
        {
            // epsneg
            return machep();
        }
    };

    template <> struct cephes_helper<double>
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE double machep()
        {
            return NumTraits<double>::epsilon() / 2;  // 1.0 - machep == 1.0
        }
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE double big() { return 1.0 / NumTraits<double>::epsilon(); }
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE double biginv()
        {
            // inverse of eps
            return NumTraits<double>::epsilon();
        }
    };

    enum IgammaComputationMode
    {
        VALUE,
        DERIVATIVE,
        SAMPLE_DERIVATIVE
    };

    template <typename Scalar> EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Scalar main_igamma_term(Scalar a, Scalar x)
    {
        /* Compute  x**a * exp(-x) / gamma(a)  */
        Scalar logax = a * numext::log(x) - x - lgamma_impl<Scalar>::run(a);
        if (logax < -numext::log(NumTraits<Scalar>::highest()) ||
            // Assuming x and a aren't Nan.
            (numext::isnan)(logax))
        {
            return Scalar(0);
        }
        return numext::exp(logax);
    }

    template <typename Scalar, IgammaComputationMode mode> EIGEN_DEVICE_FUNC int igamma_num_iterations()
    {
        /* Returns the maximum number of internal iterations for igamma computation.
   */
        if (mode == VALUE)
        {
            return 2000;
        }

        if (internal::is_same<Scalar, float>::value)
        {
            return 200;
        }
        else if (internal::is_same<Scalar, double>::value)
        {
            return 500;
        }
        else
        {
            return 2000;
        }
    }

    template <typename Scalar, IgammaComputationMode mode> struct igammac_cf_impl
    {
        /* Computes igamc(a, x) or derivative (depending on the mode)
   * using the continued fraction expansion of the complementary
   * incomplete Gamma function.
   *
   * Preconditions:
   *   a > 0
   *   x >= 1
   *   x >= a
   */
        EIGEN_DEVICE_FUNC
        static Scalar run(Scalar a, Scalar x)
        {
            const Scalar zero = 0;
            const Scalar one = 1;
            const Scalar two = 2;
            const Scalar machep = cephes_helper<Scalar>::machep();
            const Scalar big = cephes_helper<Scalar>::big();
            const Scalar biginv = cephes_helper<Scalar>::biginv();

            if ((numext::isinf)(x))
            {
                return zero;
            }

            Scalar ax = main_igamma_term<Scalar>(a, x);
            // This is independent of mode. If this value is zero,
            // then the function value is zero. If the function value is zero,
            // then we are in a neighborhood where the function value evalutes to zero,
            // so the derivative is zero.
            if (ax == zero)
            {
                return zero;
            }

            // continued fraction
            Scalar y = one - a;
            Scalar z = x + y + one;
            Scalar c = zero;
            Scalar pkm2 = one;
            Scalar qkm2 = x;
            Scalar pkm1 = x + one;
            Scalar qkm1 = z * x;
            Scalar ans = pkm1 / qkm1;

            Scalar dpkm2_da = zero;
            Scalar dqkm2_da = zero;
            Scalar dpkm1_da = zero;
            Scalar dqkm1_da = -x;
            Scalar dans_da = (dpkm1_da - ans * dqkm1_da) / qkm1;

            for (int i = 0; i < igamma_num_iterations<Scalar, mode>(); i++)
            {
                c += one;
                y += one;
                z += two;

                Scalar yc = y * c;
                Scalar pk = pkm1 * z - pkm2 * yc;
                Scalar qk = qkm1 * z - qkm2 * yc;

                Scalar dpk_da = dpkm1_da * z - pkm1 - dpkm2_da * yc + pkm2 * c;
                Scalar dqk_da = dqkm1_da * z - qkm1 - dqkm2_da * yc + qkm2 * c;

                if (qk != zero)
                {
                    Scalar ans_prev = ans;
                    ans = pk / qk;

                    Scalar dans_da_prev = dans_da;
                    dans_da = (dpk_da - ans * dqk_da) / qk;

                    if (mode == VALUE)
                    {
                        if (numext::abs(ans_prev - ans) <= machep * numext::abs(ans))
                        {
                            break;
                        }
                    }
                    else
                    {
                        if (numext::abs(dans_da - dans_da_prev) <= machep)
                        {
                            break;
                        }
                    }
                }

                pkm2 = pkm1;
                pkm1 = pk;
                qkm2 = qkm1;
                qkm1 = qk;

                dpkm2_da = dpkm1_da;
                dpkm1_da = dpk_da;
                dqkm2_da = dqkm1_da;
                dqkm1_da = dqk_da;

                if (numext::abs(pk) > big)
                {
                    pkm2 *= biginv;
                    pkm1 *= biginv;
                    qkm2 *= biginv;
                    qkm1 *= biginv;

                    dpkm2_da *= biginv;
                    dpkm1_da *= biginv;
                    dqkm2_da *= biginv;
                    dqkm1_da *= biginv;
                }
            }

            /* Compute  x**a * exp(-x) / gamma(a)  */
            Scalar dlogax_da = numext::log(x) - digamma_impl<Scalar>::run(a);
            Scalar dax_da = ax * dlogax_da;

            switch (mode)
            {
            case VALUE:
                return ans * ax;
            case DERIVATIVE:
                return ans * dax_da + dans_da * ax;
            case SAMPLE_DERIVATIVE:
            default:  // this is needed to suppress clang warning
                return -(dans_da + ans * dlogax_da) * x;
            }
        }
    };

    template <typename Scalar, IgammaComputationMode mode> struct igamma_series_impl
    {
        /* Computes igam(a, x) or its derivative (depending on the mode)
   * using the series expansion of the incomplete Gamma function.
   *
   * Preconditions:
   *   x > 0
   *   a > 0
   *   !(x > 1 && x > a)
   */
        EIGEN_DEVICE_FUNC
        static Scalar run(Scalar a, Scalar x)
        {
            const Scalar zero = 0;
            const Scalar one = 1;
            const Scalar machep = cephes_helper<Scalar>::machep();

            Scalar ax = main_igamma_term<Scalar>(a, x);

            // This is independent of mode. If this value is zero,
            // then the function value is zero. If the function value is zero,
            // then we are in a neighborhood where the function value evalutes to zero,
            // so the derivative is zero.
            if (ax == zero)
            {
                return zero;
            }

            ax /= a;

            /* power series */
            Scalar r = a;
            Scalar c = one;
            Scalar ans = one;

            Scalar dc_da = zero;
            Scalar dans_da = zero;

            for (int i = 0; i < igamma_num_iterations<Scalar, mode>(); i++)
            {
                r += one;
                Scalar term = x / r;
                Scalar dterm_da = -x / (r * r);
                dc_da = term * dc_da + dterm_da * c;
                dans_da += dc_da;
                c *= term;
                ans += c;

                if (mode == VALUE)
                {
                    if (c <= machep * ans)
                    {
                        break;
                    }
                }
                else
                {
                    if (numext::abs(dc_da) <= machep * numext::abs(dans_da))
                    {
                        break;
                    }
                }
            }

            Scalar dlogax_da = numext::log(x) - digamma_impl<Scalar>::run(a + one);
            Scalar dax_da = ax * dlogax_da;

            switch (mode)
            {
            case VALUE:
                return ans * ax;
            case DERIVATIVE:
                return ans * dax_da + dans_da * ax;
            case SAMPLE_DERIVATIVE:
            default:  // this is needed to suppress clang warning
                return -(dans_da + ans * dlogax_da) * x / a;
            }
        }
    };

#if !EIGEN_HAS_C99_MATH

    template <typename Scalar> struct igammac_impl
    {
        EIGEN_DEVICE_FUNC
        static Scalar run(Scalar a, Scalar x)
        {
            EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false), THIS_TYPE_IS_NOT_SUPPORTED);
            return Scalar(0);
        }
    };

#else

    template <typename Scalar> struct igammac_impl
    {
        EIGEN_DEVICE_FUNC
        static Scalar run(Scalar a, Scalar x)
        {
            /*  igamc()
     *
     *	Incomplete gamma integral (modified for Eigen)
     *
     *
     *
     * SYNOPSIS:
     *
     * double a, x, y, igamc();
     *
     * y = igamc( a, x );
     *
     * DESCRIPTION:
     *
     * The function is defined by
     *
     *
     *  igamc(a,x)   =   1 - igam(a,x)
     *
     *                            inf.
     *                              -
     *                     1       | |  -t  a-1
     *               =   -----     |   e   t   dt.
     *                    -      | |
     *                   | (a)    -
     *                             x
     *
     *
     * In this implementation both arguments must be positive.
     * The integral is evaluated by either a power series or
     * continued fraction expansion, depending on the relative
     * values of a and x.
     *
     * ACCURACY (float):
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      0,30        30000       7.8e-6      5.9e-7
     *
     *
     * ACCURACY (double):
     *
     * Tested at random a, x.
     *                a         x                      Relative error:
     * arithmetic   domain   domain     # trials      peak         rms
     *    IEEE     0.5,100   0,100      200000       1.9e-14     1.7e-15
     *    IEEE     0.01,0.5  0,100      200000       1.4e-13     1.6e-15
     *
     */
            /*
      Cephes Math Library Release 2.2: June, 1992
      Copyright 1985, 1987, 1992 by Stephen L. Moshier
      Direct inquiries to 30 Frost Street, Cambridge, MA 02140
    */
            const Scalar zero = 0;
            const Scalar one = 1;
            const Scalar nan = NumTraits<Scalar>::quiet_NaN();

            if ((x < zero) || (a <= zero))
            {
                // domain error
                return nan;
            }

            if ((numext::isnan)(a) || (numext::isnan)(x))
            {  // propagate nans
                return nan;
            }

            if ((x < one) || (x < a))
            {
                return (one - igamma_series_impl<Scalar, VALUE>::run(a, x));
            }

            return igammac_cf_impl<Scalar, VALUE>::run(a, x);
        }
    };

#endif  // EIGEN_HAS_C99_MATH

    /************************************************************************************************
 * Implementation of igamma (incomplete gamma integral), based on Cephes but requires C++11/C99 *
 ************************************************************************************************/

#if !EIGEN_HAS_C99_MATH

    template <typename Scalar, IgammaComputationMode mode> struct igamma_generic_impl
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE Scalar run(Scalar a, Scalar x)
        {
            EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false), THIS_TYPE_IS_NOT_SUPPORTED);
            return Scalar(0);
        }
    };

#else

    template <typename Scalar, IgammaComputationMode mode> struct igamma_generic_impl
    {
        EIGEN_DEVICE_FUNC
        static Scalar run(Scalar a, Scalar x)
        {
            /* Depending on the mode, returns
     * - VALUE: incomplete Gamma function igamma(a, x)
     * - DERIVATIVE: derivative of incomplete Gamma function d/da igamma(a, x)
     * - SAMPLE_DERIVATIVE: implicit derivative of a Gamma random variable
     * x ~ Gamma(x | a, 1), dx/da = -1 / Gamma(x | a, 1) * d igamma(a, x) / dx
     *
     * Derivatives are implemented by forward-mode differentiation.
     */
            const Scalar zero = 0;
            const Scalar one = 1;
            const Scalar nan = NumTraits<Scalar>::quiet_NaN();

            if (x == zero)
                return zero;

            if ((x < zero) || (a <= zero))
            {  // domain error
                return nan;
            }

            if ((numext::isnan)(a) || (numext::isnan)(x))
            {  // propagate nans
                return nan;
            }

            if ((x > one) && (x > a))
            {
                Scalar ret = igammac_cf_impl<Scalar, mode>::run(a, x);
                if (mode == VALUE)
                {
                    return one - ret;
                }
                else
                {
                    return -ret;
                }
            }

            return igamma_series_impl<Scalar, mode>::run(a, x);
        }
    };

#endif  // EIGEN_HAS_C99_MATH

    template <typename Scalar> struct igamma_retval
    {
        typedef Scalar type;
    };

    template <typename Scalar> struct igamma_impl : igamma_generic_impl<Scalar, VALUE>
    {
        /* igam()
   * Incomplete gamma integral.
   *
   * The CDF of Gamma(a, 1) random variable at the point x.
   *
   * Accuracy estimation. For each a in [10^-2, 10^-1...10^3] we sample
   * 50 Gamma random variables x ~ Gamma(x | a, 1), a total of 300 points.
   * The ground truth is computed by mpmath. Mean absolute error:
   * float: 1.26713e-05
   * double: 2.33606e-12
   *
   * Cephes documentation below.
   *
   * SYNOPSIS:
   *
   * double a, x, y, igam();
   *
   * y = igam( a, x );
   *
   * DESCRIPTION:
   *
   * The function is defined by
   *
   *                           x
   *                            -
   *                   1       | |  -t  a-1
   *  igam(a,x)  =   -----     |   e   t   dt.
   *                  -      | |
   *                 | (a)    -
   *                           0
   *
   *
   * In this implementation both arguments must be positive.
   * The integral is evaluated by either a power series or
   * continued fraction expansion, depending on the relative
   * values of a and x.
   *
   * ACCURACY (double):
   *
   *                      Relative error:
   * arithmetic   domain     # trials      peak         rms
   *    IEEE      0,30       200000       3.6e-14     2.9e-15
   *    IEEE      0,100      300000       9.9e-14     1.5e-14
   *
   *
   * ACCURACY (float):
   *
   *                      Relative error:
   * arithmetic   domain     # trials      peak         rms
   *    IEEE      0,30        20000       7.8e-6      5.9e-7
   *
   */
        /*
    Cephes Math Library Release 2.2: June, 1992
    Copyright 1985, 1987, 1992 by Stephen L. Moshier
    Direct inquiries to 30 Frost Street, Cambridge, MA 02140
  */

        /* left tail of incomplete gamma function:
   *
   *          inf.      k
   *   a  -x   -       x
   *  x  e     >   ----------
   *           -     -
   *          k=0   | (a+k+1)
   *
   */
    };

    template <typename Scalar> struct igamma_der_a_retval : igamma_retval<Scalar>
    {
    };

    template <typename Scalar> struct igamma_der_a_impl : igamma_generic_impl<Scalar, DERIVATIVE>
    {
        /* Derivative of the incomplete Gamma function with respect to a.
   *
   * Computes d/da igamma(a, x) by forward differentiation of the igamma code.
   *
   * Accuracy estimation. For each a in [10^-2, 10^-1...10^3] we sample
   * 50 Gamma random variables x ~ Gamma(x | a, 1), a total of 300 points.
   * The ground truth is computed by mpmath. Mean absolute error:
   * float: 6.17992e-07
   * double: 4.60453e-12
   *
   * Reference:
   * R. Moore. "Algorithm AS 187: Derivatives of the incomplete gamma
   * integral". Journal of the Royal Statistical Society. 1982
   */
    };

    template <typename Scalar> struct gamma_sample_der_alpha_retval : igamma_retval<Scalar>
    {
    };

    template <typename Scalar> struct gamma_sample_der_alpha_impl : igamma_generic_impl<Scalar, SAMPLE_DERIVATIVE>
    {
        /* Derivative of a Gamma random variable sample with respect to alpha.
   *
   * Consider a sample of a Gamma random variable with the concentration
   * parameter alpha: sample ~ Gamma(alpha, 1). The reparameterization
   * derivative that we want to compute is dsample / dalpha =
   * d igammainv(alpha, u) / dalpha, where u = igamma(alpha, sample).
   * However, this formula is numerically unstable and expensive, so instead
   * we use implicit differentiation:
   *
   * igamma(alpha, sample) = u, where u ~ Uniform(0, 1).
   * Apply d / dalpha to both sides:
   * d igamma(alpha, sample) / dalpha
   *     + d igamma(alpha, sample) / dsample * dsample/dalpha  = 0
   * d igamma(alpha, sample) / dalpha
   *     + Gamma(sample | alpha, 1) dsample / dalpha = 0
   * dsample/dalpha = - (d igamma(alpha, sample) / dalpha)
   *                   / Gamma(sample | alpha, 1)
   *
   * Here Gamma(sample | alpha, 1) is the PDF of the Gamma distribution
   * (note that the derivative of the CDF w.r.t. sample is the PDF).
   * See the reference below for more details.
   *
   * The derivative of igamma(alpha, sample) is computed by forward
   * differentiation of the igamma code. Division by the Gamma PDF is performed
   * in the same code, increasing the accuracy and speed due to cancellation
   * of some terms.
   *
   * Accuracy estimation. For each alpha in [10^-2, 10^-1...10^3] we sample
   * 50 Gamma random variables sample ~ Gamma(sample | alpha, 1), a total of 300
   * points. The ground truth is computed by mpmath. Mean absolute error:
   * float: 2.1686e-06
   * double: 1.4774e-12
   *
   * Reference:
   * M. Figurnov, S. Mohamed, A. Mnih "Implicit Reparameterization Gradients".
   * 2018
   */
    };

    /*****************************************************************************
 * Implementation of Riemann zeta function of two arguments, based on Cephes *
 *****************************************************************************/

    template <typename Scalar> struct zeta_retval
    {
        typedef Scalar type;
    };

    template <typename Scalar> struct zeta_impl_series
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE Scalar run(const Scalar)
        {
            EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false), THIS_TYPE_IS_NOT_SUPPORTED);
            return Scalar(0);
        }
    };

    template <> struct zeta_impl_series<float>
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE bool run(float& a, float& b, float& s, const float x, const float machep)
        {
            int i = 0;
            while (i < 9)
            {
                i += 1;
                a += 1.0f;
                b = numext::pow(a, -x);
                s += b;
                if (numext::abs(b / s) < machep)
                    return true;
            }

            //Return whether we are done
            return false;
        }
    };

    template <> struct zeta_impl_series<double>
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE bool run(double& a, double& b, double& s, const double x, const double machep)
        {
            int i = 0;
            while ((i < 9) || (a <= 9.0))
            {
                i += 1;
                a += 1.0;
                b = numext::pow(a, -x);
                s += b;
                if (numext::abs(b / s) < machep)
                    return true;
            }

            //Return whether we are done
            return false;
        }
    };

    template <typename Scalar> struct zeta_impl
    {
        EIGEN_DEVICE_FUNC
        static Scalar run(Scalar x, Scalar q)
        {
            /*							zeta.c
         *
         *	Riemann zeta function of two arguments
         *
         *
         *
         * SYNOPSIS:
         *
         * double x, q, y, zeta();
         *
         * y = zeta( x, q );
         *
         *
         *
         * DESCRIPTION:
         *
         *
         *
         *                 inf.
         *                  -        -x
         *   zeta(x,q)  =   >   (k+q)
         *                  -
         *                 k=0
         *
         * where x > 1 and q is not a negative integer or zero.
         * The Euler-Maclaurin summation formula is used to obtain
         * the expansion
         *
         *                n
         *                -       -x
         * zeta(x,q)  =   >  (k+q)
         *                -
         *               k=1
         *
         *           1-x                 inf.  B   x(x+1)...(x+2j)
         *      (n+q)           1         -     2j
         *  +  ---------  -  -------  +   >    --------------------
         *        x-1              x      -                   x+2j+1
         *                   2(n+q)      j=1       (2j)! (n+q)
         *
         * where the B2j are Bernoulli numbers.  Note that (see zetac.c)
         * zeta(x,1) = zetac(x) + 1.
         *
         *
         *
         * ACCURACY:
         *
         * Relative error for single precision:
         * arithmetic   domain     # trials      peak         rms
         *    IEEE      0,25        10000       6.9e-7      1.0e-7
         *
         * Large arguments may produce underflow in powf(), in which
         * case the results are inaccurate.
         *
         * REFERENCE:
         *
         * Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals,
         * Series, and Products, p. 1073; Academic Press, 1980.
         *
         */

            int i;
            Scalar p, r, a, b, k, s, t, w;

            const Scalar A[] = {
                Scalar(12.0),
                Scalar(-720.0),
                Scalar(30240.0),
                Scalar(-1209600.0),
                Scalar(47900160.0),
                Scalar(-1.8924375803183791606e9), /*1.307674368e12/691*/
                Scalar(7.47242496e10),
                Scalar(-2.950130727918164224e12),  /*1.067062284288e16/3617*/
                Scalar(1.1646782814350067249e14),  /*5.109094217170944e18/43867*/
                Scalar(-4.5979787224074726105e15), /*8.028576626982912e20/174611*/
                Scalar(1.8152105401943546773e17),  /*1.5511210043330985984e23/854513*/
                Scalar(-7.1661652561756670113e18)  /*1.6938241367317436694528e27/236364091*/
            };

            const Scalar maxnum = NumTraits<Scalar>::infinity();
            const Scalar zero = 0.0, half = 0.5, one = 1.0;
            const Scalar machep = cephes_helper<Scalar>::machep();
            const Scalar nan = NumTraits<Scalar>::quiet_NaN();

            if (x == one)
                return maxnum;

            if (x < one)
            {
                return nan;
            }

            if (q <= zero)
            {
                if (q == numext::floor(q))
                {
                    if (x == numext::floor(x) && long(x) % 2 == 0)
                    {
                        return maxnum;
                    }
                    else
                    {
                        return nan;
                    }
                }
                p = x;
                r = numext::floor(p);
                if (p != r)
                    return nan;
            }

            /* Permit negative q but continue sum until n+q > +9 .
         * This case should be handled by a reflection formula.
         * If q<0 and x is an integer, there is a relation to
         * the polygamma function.
         */
            s = numext::pow(q, -x);
            a = q;
            b = zero;
            // Run the summation in a helper function that is specific to the floating precision
            if (zeta_impl_series<Scalar>::run(a, b, s, x, machep))
            {
                return s;
            }

            w = a;
            s += b * w / (x - one);
            s -= half * b;
            a = one;
            k = zero;
            for (i = 0; i < 12; i++)
            {
                a *= x + k;
                b /= w;
                t = a * b / A[i];
                s = s + t;
                t = numext::abs(t / s);
                if (t < machep)
                {
                    break;
                }
                k += one;
                a *= x + k;
                b /= w;
                k += one;
            }
            return s;
        }
    };

    /****************************************************************************
 * Implementation of polygamma function, requires C++11/C99                 *
 ****************************************************************************/

    template <typename Scalar> struct polygamma_retval
    {
        typedef Scalar type;
    };

#if !EIGEN_HAS_C99_MATH

    template <typename Scalar> struct polygamma_impl
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE Scalar run(Scalar n, Scalar x)
        {
            EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false), THIS_TYPE_IS_NOT_SUPPORTED);
            return Scalar(0);
        }
    };

#else

    template <typename Scalar> struct polygamma_impl
    {
        EIGEN_DEVICE_FUNC
        static Scalar run(Scalar n, Scalar x)
        {
            Scalar zero = 0.0, one = 1.0;
            Scalar nplus = n + one;
            const Scalar nan = NumTraits<Scalar>::quiet_NaN();

            // Check that n is a non-negative integer
            if (numext::floor(n) != n || n < zero)
            {
                return nan;
            }
            // Just return the digamma function for n = 0
            else if (n == zero)
            {
                return digamma_impl<Scalar>::run(x);
            }
            // Use the same implementation as scipy
            else
            {
                Scalar factorial = numext::exp(lgamma_impl<Scalar>::run(nplus));
                return numext::pow(-one, nplus) * factorial * zeta_impl<Scalar>::run(nplus, x);
            }
        }
    };

#endif  // EIGEN_HAS_C99_MATH

    /************************************************************************************************
 * Implementation of betainc (incomplete beta integral), based on Cephes but requires C++11/C99 *
 ************************************************************************************************/

    template <typename Scalar> struct betainc_retval
    {
        typedef Scalar type;
    };

#if !EIGEN_HAS_C99_MATH

    template <typename Scalar> struct betainc_impl
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE Scalar run(Scalar a, Scalar b, Scalar x)
        {
            EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false), THIS_TYPE_IS_NOT_SUPPORTED);
            return Scalar(0);
        }
    };

#else

    template <typename Scalar> struct betainc_impl
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE Scalar run(Scalar, Scalar, Scalar)
        {
            /*	betaincf.c
     *
     *	Incomplete beta integral
     *
     *
     * SYNOPSIS:
     *
     * float a, b, x, y, betaincf();
     *
     * y = betaincf( a, b, x );
     *
     *
     * DESCRIPTION:
     *
     * Returns incomplete beta integral of the arguments, evaluated
     * from zero to x.  The function is defined as
     *
     *                  x
     *     -            -
     *    | (a+b)      | |  a-1     b-1
     *  -----------    |   t   (1-t)   dt.
     *   -     -     | |
     *  | (a) | (b)   -
     *                 0
     *
     * The domain of definition is 0 <= x <= 1.  In this
     * implementation a and b are restricted to positive values.
     * The integral from x to 1 may be obtained by the symmetry
     * relation
     *
     *    1 - betainc( a, b, x )  =  betainc( b, a, 1-x ).
     *
     * The integral is evaluated by a continued fraction expansion.
     * If a < 1, the function calls itself recursively after a
     * transformation to increase a to a+1.
     *
     * ACCURACY (float):
     *
     * Tested at random points (a,b,x) with a and b in the indicated
     * interval and x between 0 and 1.
     *
     * arithmetic   domain     # trials      peak         rms
     * Relative error:
     *    IEEE       0,30       10000       3.7e-5      5.1e-6
     *    IEEE       0,100      10000       1.7e-4      2.5e-5
     * The useful domain for relative error is limited by underflow
     * of the single precision exponential function.
     * Absolute error:
     *    IEEE       0,30      100000       2.2e-5      9.6e-7
     *    IEEE       0,100      10000       6.5e-5      3.7e-6
     *
     * Larger errors may occur for extreme ratios of a and b.
     *
     * ACCURACY (double):
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      0,5         10000       6.9e-15     4.5e-16
     *    IEEE      0,85       250000       2.2e-13     1.7e-14
     *    IEEE      0,1000      30000       5.3e-12     6.3e-13
     *    IEEE      0,10000    250000       9.3e-11     7.1e-12
     *    IEEE      0,100000    10000       8.7e-10     4.8e-11
     * Outputs smaller than the IEEE gradual underflow threshold
     * were excluded from these statistics.
     *
     * ERROR MESSAGES:
     *   message         condition      value returned
     * incbet domain      x<0, x>1          nan
     * incbet underflow                     nan
     */

            EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false), THIS_TYPE_IS_NOT_SUPPORTED);
            return Scalar(0);
        }
    };

    /* Continued fraction expansion #1 for incomplete beta integral (small_branch = True)
 * Continued fraction expansion #2 for incomplete beta integral (small_branch = False)
 */
    template <typename Scalar> struct incbeta_cfe
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE Scalar run(Scalar a, Scalar b, Scalar x, bool small_branch)
        {
            EIGEN_STATIC_ASSERT((internal::is_same<Scalar, float>::value || internal::is_same<Scalar, double>::value), THIS_TYPE_IS_NOT_SUPPORTED);
            const Scalar big = cephes_helper<Scalar>::big();
            const Scalar machep = cephes_helper<Scalar>::machep();
            const Scalar biginv = cephes_helper<Scalar>::biginv();

            const Scalar zero = 0;
            const Scalar one = 1;
            const Scalar two = 2;

            Scalar xk, pk, pkm1, pkm2, qk, qkm1, qkm2;
            Scalar k1, k2, k3, k4, k5, k6, k7, k8, k26update;
            Scalar ans;
            int n;

            const int num_iters = (internal::is_same<Scalar, float>::value) ? 100 : 300;
            const Scalar thresh = (internal::is_same<Scalar, float>::value) ? machep : Scalar(3) * machep;
            Scalar r = (internal::is_same<Scalar, float>::value) ? zero : one;

            if (small_branch)
            {
                k1 = a;
                k2 = a + b;
                k3 = a;
                k4 = a + one;
                k5 = one;
                k6 = b - one;
                k7 = k4;
                k8 = a + two;
                k26update = one;
            }
            else
            {
                k1 = a;
                k2 = b - one;
                k3 = a;
                k4 = a + one;
                k5 = one;
                k6 = a + b;
                k7 = a + one;
                k8 = a + two;
                k26update = -one;
                x = x / (one - x);
            }

            pkm2 = zero;
            qkm2 = one;
            pkm1 = one;
            qkm1 = one;
            ans = one;
            n = 0;

            do
            {
                xk = -(x * k1 * k2) / (k3 * k4);
                pk = pkm1 + pkm2 * xk;
                qk = qkm1 + qkm2 * xk;
                pkm2 = pkm1;
                pkm1 = pk;
                qkm2 = qkm1;
                qkm1 = qk;

                xk = (x * k5 * k6) / (k7 * k8);
                pk = pkm1 + pkm2 * xk;
                qk = qkm1 + qkm2 * xk;
                pkm2 = pkm1;
                pkm1 = pk;
                qkm2 = qkm1;
                qkm1 = qk;

                if (qk != zero)
                {
                    r = pk / qk;
                    if (numext::abs(ans - r) < numext::abs(r) * thresh)
                    {
                        return r;
                    }
                    ans = r;
                }

                k1 += one;
                k2 += k26update;
                k3 += two;
                k4 += two;
                k5 += one;
                k6 -= k26update;
                k7 += two;
                k8 += two;

                if ((numext::abs(qk) + numext::abs(pk)) > big)
                {
                    pkm2 *= biginv;
                    pkm1 *= biginv;
                    qkm2 *= biginv;
                    qkm1 *= biginv;
                }
                if ((numext::abs(qk) < biginv) || (numext::abs(pk) < biginv))
                {
                    pkm2 *= big;
                    pkm1 *= big;
                    qkm2 *= big;
                    qkm1 *= big;
                }
            } while (++n < num_iters);

            return ans;
        }
    };

    /* Helper functions depending on the Scalar type */
    template <typename Scalar> struct betainc_helper
    {
    };

    template <> struct betainc_helper<float>
    {
        /* Core implementation, assumes a large (> 1.0) */
        EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE float incbsa(float aa, float bb, float xx)
        {
            float ans, a, b, t, x, onemx;
            bool reversed_a_b = false;

            onemx = 1.0f - xx;

            /* see if x is greater than the mean */
            if (xx > (aa / (aa + bb)))
            {
                reversed_a_b = true;
                a = bb;
                b = aa;
                t = xx;
                x = onemx;
            }
            else
            {
                a = aa;
                b = bb;
                t = onemx;
                x = xx;
            }

            /* Choose expansion for optimal convergence */
            if (b > 10.0f)
            {
                if (numext::abs(b * x / a) < 0.3f)
                {
                    t = betainc_helper<float>::incbps(a, b, x);
                    if (reversed_a_b)
                        t = 1.0f - t;
                    return t;
                }
            }

            ans = x * (a + b - 2.0f) / (a - 1.0f);
            if (ans < 1.0f)
            {
                ans = incbeta_cfe<float>::run(a, b, x, true /* small_branch */);
                t = b * numext::log(t);
            }
            else
            {
                ans = incbeta_cfe<float>::run(a, b, x, false /* small_branch */);
                t = (b - 1.0f) * numext::log(t);
            }

            t += a * numext::log(x) + lgamma_impl<float>::run(a + b) - lgamma_impl<float>::run(a) - lgamma_impl<float>::run(b);
            t += numext::log(ans / a);
            t = numext::exp(t);

            if (reversed_a_b)
                t = 1.0f - t;
            return t;
        }

        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE float incbps(float a, float b, float x)
        {
            float t, u, y, s;
            const float machep = cephes_helper<float>::machep();

            y = a * numext::log(x) + (b - 1.0f) * numext::log1p(-x) - numext::log(a);
            y -= lgamma_impl<float>::run(a) + lgamma_impl<float>::run(b);
            y += lgamma_impl<float>::run(a + b);

            t = x / (1.0f - x);
            s = 0.0f;
            u = 1.0f;
            do
            {
                b -= 1.0f;
                if (b == 0.0f)
                {
                    break;
                }
                a += 1.0f;
                u *= t * b / a;
                s += u;
            } while (numext::abs(u) > machep);

            return numext::exp(y) * (1.0f + s);
        }
    };

    template <> struct betainc_impl<float>
    {
        EIGEN_DEVICE_FUNC
        static float run(float a, float b, float x)
        {
            const float nan = NumTraits<float>::quiet_NaN();
            float ans, t;

            if (a <= 0.0f)
                return nan;
            if (b <= 0.0f)
                return nan;
            if ((x <= 0.0f) || (x >= 1.0f))
            {
                if (x == 0.0f)
                    return 0.0f;
                if (x == 1.0f)
                    return 1.0f;
                // mtherr("betaincf", DOMAIN);
                return nan;
            }

            /* transformation for small aa */
            if (a <= 1.0f)
            {
                ans = betainc_helper<float>::incbsa(a + 1.0f, b, x);
                t = a * numext::log(x) + b * numext::log1p(-x) + lgamma_impl<float>::run(a + b) - lgamma_impl<float>::run(a + 1.0f) -
                    lgamma_impl<float>::run(b);
                return (ans + numext::exp(t));
            }
            else
            {
                return betainc_helper<float>::incbsa(a, b, x);
            }
        }
    };

    template <> struct betainc_helper<double>
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_STRONG_INLINE double incbps(double a, double b, double x)
        {
            const double machep = cephes_helper<double>::machep();

            double s, t, u, v, n, t1, z, ai;

            ai = 1.0 / a;
            u = (1.0 - b) * x;
            v = u / (a + 1.0);
            t1 = v;
            t = u;
            n = 2.0;
            s = 0.0;
            z = machep * ai;
            while (numext::abs(v) > z)
            {
                u = (n - b) * x / n;
                t *= u;
                v = t / (a + n);
                s += v;
                n += 1.0;
            }
            s += t1;
            s += ai;

            u = a * numext::log(x);
            // TODO: gamma() is not directly implemented in Eigen.
            /*
    if ((a + b) < maxgam && numext::abs(u) < maxlog) {
      t = gamma(a + b) / (gamma(a) * gamma(b));
      s = s * t * pow(x, a);
    }
    */
            t = lgamma_impl<double>::run(a + b) - lgamma_impl<double>::run(a) - lgamma_impl<double>::run(b) + u + numext::log(s);
            return s = numext::exp(t);
        }
    };

    template <> struct betainc_impl<double>
    {
        EIGEN_DEVICE_FUNC
        static double run(double aa, double bb, double xx)
        {
            const double nan = NumTraits<double>::quiet_NaN();
            const double machep = cephes_helper<double>::machep();
            // const double maxgam = 171.624376956302725;

            double a, b, t, x, xc, w, y;
            bool reversed_a_b = false;

            if (aa <= 0.0 || bb <= 0.0)
            {
                return nan;  // goto domerr;
            }

            if ((xx <= 0.0) || (xx >= 1.0))
            {
                if (xx == 0.0)
                    return (0.0);
                if (xx == 1.0)
                    return (1.0);
                // mtherr("incbet", DOMAIN);
                return nan;
            }

            if ((bb * xx) <= 1.0 && xx <= 0.95)
            {
                return betainc_helper<double>::incbps(aa, bb, xx);
            }

            w = 1.0 - xx;

            /* Reverse a and b if x is greater than the mean. */
            if (xx > (aa / (aa + bb)))
            {
                reversed_a_b = true;
                a = bb;
                b = aa;
                xc = xx;
                x = w;
            }
            else
            {
                a = aa;
                b = bb;
                xc = w;
                x = xx;
            }

            if (reversed_a_b && (b * x) <= 1.0 && x <= 0.95)
            {
                t = betainc_helper<double>::incbps(a, b, x);
                if (t <= machep)
                {
                    t = 1.0 - machep;
                }
                else
                {
                    t = 1.0 - t;
                }
                return t;
            }

            /* Choose expansion for better convergence. */
            y = x * (a + b - 2.0) - (a - 1.0);
            if (y < 0.0)
            {
                w = incbeta_cfe<double>::run(a, b, x, true /* small_branch */);
            }
            else
            {
                w = incbeta_cfe<double>::run(a, b, x, false /* small_branch */) / xc;
            }

            /* Multiply w by the factor
         a      b   _             _     _
        x  (1-x)   | (a+b) / ( a | (a) | (b) ) .   */

            y = a * numext::log(x);
            t = b * numext::log(xc);
            // TODO: gamma is not directly implemented in Eigen.
            /*
    if ((a + b) < maxgam && numext::abs(y) < maxlog && numext::abs(t) < maxlog)
    {
      t = pow(xc, b);
      t *= pow(x, a);
      t /= a;
      t *= w;
      t *= gamma(a + b) / (gamma(a) * gamma(b));
    } else {
    */
            /* Resort to logarithms.  */
            y += t + lgamma_impl<double>::run(a + b) - lgamma_impl<double>::run(a) - lgamma_impl<double>::run(b);
            y += numext::log(w / a);
            t = numext::exp(y);

            /* } */
            // done:

            if (reversed_a_b)
            {
                if (t <= machep)
                {
                    t = 1.0 - machep;
                }
                else
                {
                    t = 1.0 - t;
                }
            }
            return t;
        }
    };

#endif  // EIGEN_HAS_C99_MATH

}  // end namespace internal

namespace numext {

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(lgamma, Scalar) lgamma(const Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(lgamma, Scalar)::run(x);
    }

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(digamma, Scalar) digamma(const Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(digamma, Scalar)::run(x);
    }

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(zeta, Scalar) zeta(const Scalar& x, const Scalar& q)
    {
        return EIGEN_MATHFUNC_IMPL(zeta, Scalar)::run(x, q);
    }

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(polygamma, Scalar) polygamma(const Scalar& n, const Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(polygamma, Scalar)::run(n, x);
    }

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(erf, Scalar) erf(const Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(erf, Scalar)::run(x);
    }

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(erfc, Scalar) erfc(const Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(erfc, Scalar)::run(x);
    }

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(ndtri, Scalar) ndtri(const Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(ndtri, Scalar)::run(x);
    }

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(igamma, Scalar) igamma(const Scalar& a, const Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(igamma, Scalar)::run(a, x);
    }

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(igamma_der_a, Scalar) igamma_der_a(const Scalar& a, const Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(igamma_der_a, Scalar)::run(a, x);
    }

    template <typename Scalar>
    EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(gamma_sample_der_alpha, Scalar) gamma_sample_der_alpha(const Scalar& a, const Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(gamma_sample_der_alpha, Scalar)::run(a, x);
    }

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(igammac, Scalar) igammac(const Scalar& a, const Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(igammac, Scalar)::run(a, x);
    }

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(betainc, Scalar) betainc(const Scalar& a, const Scalar& b, const Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(betainc, Scalar)::run(a, b, x);
    }

}  // end namespace numext
}  // end namespace Eigen

#endif  // EIGEN_SPECIAL_FUNCTIONS_H
